[用户互动]高中数学所有知识点

§04. 三角函数  知识要点
1. ①与（0°≤＜360°）终边相同的角的集合（角与角的终边重合）：
②终边在x轴上的角的集合：
③终边在y轴上的角的集合：
④终边在坐标轴上的角的集合：
⑤终边在y=x轴上的角的集合：
⑥终边在轴上的角的集合：
⑦若角与角的终边关于x轴对称，则角与角的关系：
⑧若角与角的终边关于y轴对称，则角与角的关系：
⑨若角与角的终边在一条直线上，则角与角的关系：
⑩角与角的终边互相垂直，则角与角[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif[/img]的关系：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.gif[/img]
2. 角度与弧度的互换关系：360°=2[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.gif[/img] 180°=[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.gif[/img] 1°=0.01745  1=57.30°=57°18′
注意：正角的弧度数为正数，负角的弧度数为负数，零角的弧度数为零.
、弧度与角度互换公式：  1rad＝[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.gif[/img]°≈57.30°=57°18ˊ．     1°＝[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image034.gif[/img]≈0.01745（rad）
3、弧长公式：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image036.gif[/img].       扇形面积公式：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image038.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image040.gif[/img]4、三角函数：设[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image042.gif[/img]是一个任意角，在[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image042.gif[/img]的终边上任取（异于原点的）一点P（x,y）P与原点的距离为r，则  [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image044.gif[/img]；  [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.gif[/img]；  [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.gif[/img]；  [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif[/img]；  [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image052.gif[/img]；. [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image054.gif[/img].
5、三角函数在各象限的符号：（一全二正弦，三切四余弦）
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image056.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image058.gif[/img]

[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image060.gif[/img]6、三角函数线
  正弦线：MP;   余弦线：OM;    正切线： AT.

7. 三角函数的定义域：

三角函数	定义域
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif[/img]sinx	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image064.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif[/img]cosx	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image064.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif[/img]tanx	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image067.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif[/img]cotx	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image069.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif[/img]secx	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image067.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif[/img]cscx	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image069.gif[/img]

8、同角三角函数的基本关系式：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image071.gif[/img]  [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image073.gif[/img]  
9、诱导公式：
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image087.gif[/img]
“奇变偶不变，符号看象限”
三角函数的公式：（一）基本关系
                                           
公式组二                  公式组三
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image089.gif[/img]                                            [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image091.gif[/img]  [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image093.gif[/img]    
公式组四               公式组五               公式组六            
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image095.gif[/img]  [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image097.gif[/img]  [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image099.gif[/img]                      
（二）角与角之间的互换


公式组一                                  公式组二
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image101.gif[/img]   [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image103.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image105.gif[/img]   [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image107.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image109.gif[/img]   [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image111.gif[/img]    
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image113.gif[/img]   [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image115.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image117.gif[/img]           [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image119.gif[/img]  
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image121.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image123.gif[/img]          
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image125.gif[/img]公式组三                    公式组四                                    公式组五
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image127.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image129.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image131.gif[/img]        
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image133.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image135.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image137.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image139.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image141.gif[/img]  

[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image143.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image145.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image147.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image149.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image151.gif[/img]    

[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image153.gif[/img],[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image155.gif[/img],[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image157.gif[/img],[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image159.gif[/img].



10. 正弦、余弦、正切、余切函数的图象的性质：

	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image161.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image163.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image165.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image167.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image169.gif[/img] （A、[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image171.gif[/img]＞0）
定义域	R	R	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image173.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image175.gif[/img]	R
值域	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image177.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image177.gif[/img]	R	R	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image179.gif[/img]
周期性	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image181.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.gif[/img]	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image183.gif[/img]
奇偶性	奇函数	偶函数	奇函数	奇函数	当[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image185.gif[/img]非奇非偶 当[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image187.gif[/img]奇函数
单调性	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image189.gif[/img]上为增函数；[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image191.gif[/img]上为减函数（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image193.gif[/img]）	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image195.gif[/img]；上为增函数[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image197.gif[/img] 上为减函数 （[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image193.gif[/img]）	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image199.gif[/img]上为增函数（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image193.gif[/img]）	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image201.gif[/img]上为减函数（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image193.gif[/img]）	[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image203.gif[/img]上为增函数； [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image205.gif[/img]上为减函数（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image193.gif[/img]）

[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image207.gif[/img]注意：①[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image209.gif[/img]与[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image211.gif[/img]的单调性正好相反；[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image213.gif[/img]与[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image163.gif[/img]的单调性也同样相反.一般地，若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image216.gif[/img]在[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image218.gif[/img]上递增（减），则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image220.gif[/img]在[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image218.gif[/img]上递减（增）.
②[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image222.gif[/img]与[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image224.gif[/img]的周期是[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.gif[/img].
③[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image226.gif[/img]或[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image228.gif[/img]（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image230.gif[/img]）的周期[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image232.gif[/img].
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image234.gif[/img]的周期为2[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image236.gif[/img]（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image238.gif[/img]，如图，翻折无效）.
④[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image226.gif[/img]的对称轴方程是[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image240.gif[/img]（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image193.gif[/img]），对称中心（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image243.gif[/img]）；[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image228.gif[/img]的对称轴方程是[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image245.gif[/img]（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image193.gif[/img]），对称中心（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image247.gif[/img]）；[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image249.gif[/img]的对称中心（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image251.gif[/img]）.
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image253.gif[/img]
⑤当[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image255.gif[/img]·[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image257.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image259.gif[/img]；[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image255.gif[/img]·[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image261.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image263.gif[/img].
⑥[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image163.gif[/img]与[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image265.gif[/img]是同一函数,而[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image267.gif[/img]是偶函数，则
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image269.gif[/img].
⑦函数[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image271.gif[/img]在[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image273.gif[/img]上为增函数.（×） [只能在某个单调区间单调递增. 若在整个定义域，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image271.gif[/img]为增函数，同样也是错误的].
⑧定义域关于原点对称是[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image276.gif[/img]具有奇偶性的必要不充分条件.（奇偶性的两个条件：一是定义域关于原点对称（奇偶都要），二是满足奇偶性条件，偶函数：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image278.gif[/img]，奇函数：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image280.gif[/img]）
奇偶性的单调性：奇同偶反. 例如：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image271.gif[/img]是奇函数，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image283.gif[/img]是非奇非偶.（定义域不关于原点对称）
奇函数特有性质：若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image285.gif[/img]的定义域，则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image276.gif[/img]一定有[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image287.gif[/img].（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image289.gif[/img]的定义域，则无此性质）
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image291.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image293.gif[/img]⑨[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image295.gif[/img]不是周期函数；[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image297.gif[/img]为周期函数（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image299.gif[/img]）；
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image301.gif[/img]是周期函数（如图）；[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image303.gif[/img]为周期函数（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image299.gif[/img]）；
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image305.gif[/img]的周期为[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image236.gif[/img]（如图），并非所有周期函数都有最小正周期，例如：
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image308.gif[/img].
⑩[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image310.gif[/img] 有[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image312.gif[/img].
11、三角函数图象的作法：
１）、几何法：
２）、描点法及其特例——五点作图法（正、余弦曲线），三点二线作图法（正、余切曲线）.
３）、利用图象变换作三角函数图象．
三角函数的图象变换有振幅变换、周期变换和相位变换等．
函数y＝Asin（ωx＋φ）的振幅|A|，周期[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image314.gif[/img]，频率[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image316.gif[/img]，相位[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image318.gif[/img]初相[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image320.gif[/img]（即当x＝0时的相位）．（当A＞0，ω＞0 时以上公式可去绝对值符号），
由y＝sinx的图象上的点的横坐标保持不变，纵坐标伸长（当|A|＞1）或缩短（当0＜|A|＜1）到原来的|A|倍，得到y＝Asinx的图象，叫做振幅变换或叫沿y轴的伸缩变换．（用y/A替换y）
由y＝sinx的图象上的点的纵坐标保持不变，横坐标伸长（0＜|ω|＜1）或缩短（|ω|＞1）到原来的[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image322.gif[/img]倍，得到y＝sinω x的图象，叫做周期变换或叫做沿x轴的伸缩变换．(用ωx替换x)
由y＝sinx的图象上所有的点向左（当φ＞0）或向右（当φ＜0）平行移动｜φ｜个单位，得到y＝sin（x＋φ）的图象，叫做相位变换或叫做沿x轴方向的平移．(用x＋φ替换x)
由y＝sinx的图象上所有的点向上（当b＞0）或向下（当b＜0）平行移动｜b｜个单位，得到y＝sinx＋b的图象叫做沿y轴方向的平移．（用y+(-b)替换y）
由y＝sinx的图象利用图象变换作函数y＝Asin（ωx＋φ）（A＞0，ω＞0）（x∈R）的图象，要特别注意：当周期变换和相位变换的先后顺序不同时，原图象延x轴量伸缩量的区别。
4、反三角函数：
函数y＝sinx，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image324.gif[/img]的反函数叫做反正弦函数，记作y＝arcsinx，它的定义域是［－1，1］，值域是[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image326.gif[/img]．
函数y＝cosx，（x∈［0，π］）的反应函数叫做反余弦函数，记作y＝arccosx，它的定义域是［－1，1］，值域是［0，π］．
函数y＝tanx，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image328.gif[/img]的反函数叫做反正切函数，记作y＝arctanx，它的定义域是（－∞，＋∞），值域是[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image330.gif[/img]．
函数y＝ctgx，［x∈（0，π）］的反函数叫做反余切函数，记作y＝arcctgx，它的定义域是（－∞，＋∞），值域是（0，π）．

II. 竞赛知识要点
一、反三角函数.
1. 反三角函数：⑴反正弦函数[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image332.gif[/img]是奇函数，故[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image334.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image336.gif[/img]（一定要注明定义域，若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image338.gif[/img]，没有[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image340.gif[/img]与[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image342.gif[/img]一一对应，故[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image344.gif[/img]无反函数）
注：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image346.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image347.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image349.gif[/img].
⑵反余弦函数[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image351.gif[/img]非奇非偶，但有[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image353.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image347.gif[/img].
注：①[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image355.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image356.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image358.gif[/img].
②[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image360.gif[/img]是偶函数，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image362.gif[/img]非奇非偶，而[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image364.gif[/img]和[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image366.gif[/img]为奇函数.
⑶反正切函数：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image368.gif[/img]，定义域[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image370.gif[/img]，值域（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image372.gif[/img]），[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image373.gif[/img]是奇函数，
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image375.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image377.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image378.gif[/img].
注：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image380.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image377.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image382.gif[/img].
⑷反余切函数：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image384.gif[/img]，定义域[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image385.gif[/img]，值域（[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image372.gif[/img]），[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image387.gif[/img]是非奇非偶.
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image389.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image377.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image390.gif[/img].
注：①[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image392.gif[/img]，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image377.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image393.gif[/img].
②[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image395.gif[/img]与[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image397.gif[/img]互为奇函数，[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image399.gif[/img]同理为奇而[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image401.gif[/img]与[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image403.gif[/img]非奇非偶但满足[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image405.gif[/img].

⑵ 正弦、余弦、正切、余切函数的解集：
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image407.gif[/img]的取值范围   解集                             [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image407.gif[/img]的取值范围   解集
①[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image409.gif[/img]的解集                               ②[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image411.gif[/img]的解集
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image413.gif[/img]＞1          [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image415.gif[/img]                              [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image416.gif[/img]＞1            [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image415.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image417.gif[/img]=1      [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image419.gif[/img]            [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image420.gif[/img]=1   [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image422.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image420.gif[/img]＜1     [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image424.gif[/img]       [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image425.gif[/img]＜1  [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image427.gif[/img]
③[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image429.gif[/img]的解集：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image431.gif[/img]    
③[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image433.gif[/img]的解集：[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image435.gif[/img]


[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image437.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image439.gif[/img][img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image441.gif[/img]二、三角恒等式.
组一

组二
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image443.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image445.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image447.gif[/img]
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image449.gif[/img]
组三 三角函数不等式
[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image451.gif[/img]＜[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image453.gif[/img]＜[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image455.gif[/img]            [img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image457.gif[/img]在[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image459.gif[/img]上是减函数
若[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image461.gif[/img]，则[img]file:///C:/Users/ADMINI~1/AppData/Local/Temp/msohtmlclip1/01/clip_image463.gif[/img]

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这是什么东西也蛮发

数学渣前来报道哈

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